%0 Journal Article
%T 贝塔函数在积分计算中的一个应用<br>An Application of Beta Function in Integral Calculation
%A 魏贺杰
%A 王馨雨
%J Pure Mathematics
%P 2307-2312
%@ 2160-7605
%D 2023
%I Hans Publishing
%R 10.12677/PM.2023.138237
%X 对于定积分的计算来说，典型的做法是先求出原函数，再利用牛顿莱布尼茨公式代入上下限进行计算。但对于一些难度较大的定积分计算问题，如果只局限于这种方法，计算过程将会非常的繁杂，甚至计算不出结果。本文借助第一类欧拉积分–贝塔函数的对称性和递推公式等性质推导出形如I(m,n)=∫<sub>0</sub><sup>π/2</sup>cos<sup>m</sup>x？sin<sup>n</sup>xdx(其中m,n∈？)的定积分的递推公式和计算公式，从而为这类特殊的定积分计算提供了一种有效的解决方法，大大简化了计算量。<br />
For the calculation of definite integrals, the classical method is to find the original function, and then use Newton’s Leibniz formula to substitute the upper and lower limits for calculation. However, for some difficult definite integral calculation problems, if it is limited to this method, the calculation process will be very complicated, and even the result cannot be calculated. By using the properties of Euler integral-beta function, the recursive and calculation formulas are induced for the problem I(m,n)=∫<sub>0</sub><sup>π/2</sup>cos<sup>m</sup>x？sin<sup>n</sup>xdx (where m, n are non-negative integers), which provides the effective method of solving some special types of definite integral calculation for us and simplifies the calculation greatly.
%K 贝塔函数，积分计算<br>Beta Function
%K Integral Calculation
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=70352